There are some words where this happens (e.g. 'mąż' - husband, 'dąb' - oak, 'ząb' -tooth), but it's not a rule and there are plenty of words where this does not happen.
My answer "The children are making many mistakes" was not accepted. Is there an issue with the verb tense "are making" as opposed to "make"?
No, it's just that it seems more logical that they do it "generally". But in some contexts of course 'are making' could also work, so added it.
Some people think that it is a 'countable vs uncountable' difference, but it is not true. They are interchangeable in almost any context.
The difference is that "dużo' only has Nominative and Accusative, and sometimes it can be used in Genitive (I guess that's rather when it's negated Accusative). "wiele" doesn't have this problem, it can be used in any case.
I'm confused - by saying dużo only has Nom. and Acc. (and Gen only when it is negated Acc.), are you referring to the case of the noun it's describing? If so, this sentence here is an example of the noun it's describing being Genitive--without being negated Accusative. And isn't that the typical case used by nouns that dużo describes?
Well, that's what I've read, because as a native I am not really... aware of it ;) Let's look at it more closely:
Nominative: "Dużo dzieci robi błędy" -> no problem at all.
Accusative: "Widzę dużo dzieci" -> no problem at all.
Genitive... that's the part that confused me a bit, but now I think I get it. I understand that it can be used when it's needed by the verb, but not if it is needed by a preposition.
"Potrzebuję dużo wody" -> no problem. Although "Szukam dużo wody" sounds wrong to me, as would "Szukam wiele wody"... I guess you'd have to say "Szukam dużej ilości wody" (a big amount of water) :/
But "Chodzę do dużo sklepów" - that's clearly wrong. "Chodzę do wielu sklepów" works. So this is the situation when I can't use Genitive of 'dużo' when this Genitive is caused by the preposition.
And for the other cases - also, you cannot really use 'dużo', it will sound wrong.
Oh, and for your last sentence - yes, 'dużo' needs Genitive, but we're discussing what case 'dużo' itself takes.